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Unformatted text preview: 1 Solutions 81 44. By exercise 43 this is true for
. Now apply induction. If is given
and we assume the result is true for derivatives of order
, then
but not in
. Another application of exercise 43 then shows that
but not in
.
45. First note the following trigonometric identities: Thus if and
are two basic trig functions (
or
) then
is a linear combination of basic trig functions. Now if
and
then
.
Since
is a linear combinations of basic trig functions then
is a linear combination of exponential polynomial. Now if
and
is any exponential polynomial then they are each made up of linear combinations of simple exponential polynomials and their product is a sum
of terms of products of simple exponential polynomials, which we have
shown is the sum of possibly two simple exponential polynomials. It now
follows that
is an exponential polynomial.
46. Observe that
. So the translate of an exponential function
is a multiple of an exponential function. Also, if
is a polynomial so is
. By the addition rule for
we have
and similarly for . It follows that all these translates remain
in . By Theorem ?? and Exercise ?? the result follows.
47. Since is a linear combination of terms of the form
and
it is enough to show that the derivative of each of these terms
is again in . But, by the product rule we have a linear combination of simple exponential polynomials and hence back in
. A similar calculation holds for
.
48. All exponential polynomials are continuous functions on . If
then
is an exponential polynomial. However,
is not
continuous at . Thus
is not an exponential polynomial. Section 2.6
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.
 Fall '08
 BELL,D

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